These
simple formulas use the formula for field
due to a current loop to
obtain the magnetic field at any point along the axis of
an ideal Helmholtz coil.

Ideal
Helmholtz coil in cross section view.

The ideal Helmholtz coil consists of two
coaxial circular current loops with the same radius,
separated from each other by one radius. In other words,
the loops are l apart, such that l = r.

Bx
is the magnetic field, in teslas, at any point on the
axis of the Helmholtz coil. The direction of the field is
perpendicular to the plane of the loops.

m0 is the
permeability constant (1.26x10-6 H/m)

i is the current in
the wire, in amperes.

r is the radius of
the current loops, in meters.

gis the ratio, x/r, where x
is the distance, on axis, from the center of the
Helmholtz coil, and r is the radius of the coil.

Note: different units for
r may be used as long as the permeability constant is
correct for that unit system.

Special
Case: x = 0

Bo
is the magnetic field, in teslas, at the center of the
Helmholtz coil. The direction of the field is
perpendicular to the plane of the loops.

The
following graph demonstrates the superior central field
uniformity of an ideal Helmholtz coil when compared to a
solenoid of the same aspect ratio (that is, a solenoid
which is as long as its radius):

I am developing an updated and expanded
version of the magnet web site and encourage you to
visit this work in progress
and comment on what you see. As a self-employed software engineer
trying to balance time between work and magnets, your encouragement
in building this site is always welcome!

Please consider clicking the PayPal link and making a contribution
to support the maintenance and development of this site. Thank you
for visiting!